Marked nodal curves with vector fields
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arXiv
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866915048238088192 |
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| author | Zahariuc, Adrian |
| author_facet | Zahariuc, Adrian |
| contents | We discuss two operations on nodal curves with (logarithmic) vector fields, which resemble the `stabilization' construction in Knudsen's proof that $\bar{\mathcal M}_{g,n+1}$ is the universal curve over $\bar{\mathcal M}_{g,n}$. We prove that both operations work in families (commute with base change). We construct inverse operations under suitable assumptions, which allow us to prove a technical result quite similar to Knudsen's, in the case of curves with vector fields.
As an application, we prove that the Losev--Manin compactification of the space of configurations of $n$ points on ${\mathbb P}^1 \backslash \{0,\infty\}$ modulo scaling degenerates isotrivially to a compactification of the space of configurations of $n$ points on ${\mathbb A}^1$ modulo translation, and the natural group actions fit together globally. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_13743 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Marked nodal curves with vector fields Zahariuc, Adrian Algebraic Geometry 14H10, 14M27 We discuss two operations on nodal curves with (logarithmic) vector fields, which resemble the `stabilization' construction in Knudsen's proof that $\bar{\mathcal M}_{g,n+1}$ is the universal curve over $\bar{\mathcal M}_{g,n}$. We prove that both operations work in families (commute with base change). We construct inverse operations under suitable assumptions, which allow us to prove a technical result quite similar to Knudsen's, in the case of curves with vector fields. As an application, we prove that the Losev--Manin compactification of the space of configurations of $n$ points on ${\mathbb P}^1 \backslash \{0,\infty\}$ modulo scaling degenerates isotrivially to a compactification of the space of configurations of $n$ points on ${\mathbb A}^1$ modulo translation, and the natural group actions fit together globally. |
| title | Marked nodal curves with vector fields |
| topic | Algebraic Geometry 14H10, 14M27 |
| url | https://arxiv.org/abs/2111.13743 |